Derivation of the spherical polar Laplacian
Posted by peeterjoot on October 9, 2010
In  was a Geometric Algebra derivation of the 2D polar Laplacian by squaring the quadient. In  was a factorization of the spherical polar unit vectors in a tidy compact form. Here both these ideas are utilized to derive the spherical polar form for the Laplacian, an operation that is strictly algebraic (squaring the gradient) provided we operate on the unit vectors correctly.
Our rotation multivector.
Our starting point is a pair of rotations. We rotate first in the plane by
Then apply a rotation in the plane
The composition of rotations now gives us
Expressions for the unit vectors.
The unit vectors in the rotated frame can now be calculated. With we can calculate
Performing these we get
Summarizing these are
Derivatives of the unit vectors.
We’ll need the partials. Most of these can be computed from 3.9 by inspection, and are
Expanding the Laplacian.
We note that the line element is , so our gradient in spherical coordinates is
We can now evaluate the Laplacian
Evaluating these one set at a time we have
Summing these we have
This is often written with a chain rule trick to considate the and partials
It’s simple to verify that this is identical to 5.23.
 Peeter Joot. Polar form for the gradient and Laplacian. [online]. http://sites.google.com/site/peeterjoot/math2009/polarGradAndLaplacian.pdf.
 Peeter Joot. Spherical Polar unit vectors in exponential form. [online]. http://sites.google.com/site/peeterjoot/math2009/sphericalPolarUnit.pdf .